projectivity
Let $PG(V)$ and $PG(W)$ be projective geometries^{}, with $V,W$ vector spaces^{} over a field $K$. A function $p$ from $PG(V)$ to $PG(W)$ is called a projective transformation, or simply a projectivity if

1.
$p$ is a bijection^{}, and

2.
$p$ is order preserving.
A projective property is any geometric property, such as incidence, linearity, etc… that is preserved under a projectivity.
From the definition, we see that a projectivity $p$ carries 0 to 0, $V$ to $W$. Furthermore, it carries points to points, lines to lines, planes to planes, etc.. In short, $p$ preserves linearity. Because $p$ is a bijection, $p$ also preserves dimensions^{}, that is $dim(S)=dim(p(S))$, for any subspace^{} $S$ of $V$. In particular, $dim(V)=dim(W)$. Other properties preserved by $p$ are incidence: if $S\cap T\ne \mathrm{\varnothing}$, then $p(S)\cap p(T)\ne \mathrm{\varnothing}$; and cross ratios (http://planetmath.org/CrossRatio).
Every bijective^{} semilinear transformation defines a projectiviity. To see this, let $f:V\to W$ be a semilinear transformation. If $S$ is a subspace of $V$, then $f(S)$ is a subspace of $W$, as $x,y\in f(S)$, then $x+y=f(a)+f(b)=f(a+b)\in f(S)$, and $\alpha x={\beta}^{\theta}x={\beta}^{\theta}f(a)=f(\beta a)\in f(S)$, where $\theta $ is an automorphism^{} of the common underlying field $K$. Also, if $S$ is a subspace of a subspace $T$ of $V$, then $f(S)$ is a subspace of $f(T)$. Now if we define ${f}^{*}:PG(V)\to PG(W)$ by ${f}^{*}(S)=f(S)$, it is easy to see that ${f}^{*}$ is a projectivity.
Conversely, if $V$ and $W$ are of finite dimension greater than $2$, then a projectivity $p:PG(V)\to PG(W)$ induces a semilinear transformation $\widehat{p}:V\to W$. This highly nontrivial fact is the (first) fundamental theorem of projective geometry^{}.
If the semilinear transformation induced by the projectivity $p$ is in fact a linear transformation, then $p$ is a collineation^{}: three distinct collinear points are mapped to three distinct collinear points.
Remark. The definition given in this entry is a generalization^{} of the definition typically given for a projective transformation. In the more restictive definition, a projectivity $p$ is defined merely as a bijection between two projective spaces^{} that is induced by a linear isomorphism. More precisely, if $P(V)$ and $P(W)$ are projective spaces induced by the vector spaces $V$ and $W$, if $L:V\to W$ is a bijective linear transformation, then $p=P(L):P(V)\to P(W)$ defined by
$$P(L)[v]=[Lv]$$ 
is the corresponding projective transformation. $[v]$ is the homogeneous coordinate representation of $v$. In this definition, a projectiity is always a collineation. In the case where the vector spaces are finite dimensional with specified bases, $p$ is expressible in terms of an invertible matrix ($Lv=Av$ where $A$ is an invertible matrix).
Title  projectivity 
Canonical name  Projectivity 
Date of creation  20130322 15:58:00 
Last modified on  20130322 15:58:00 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 51A10 
Classification  msc 51A05 
Related topic  PolaritiesAndForms 
Related topic  SesquilinearFormsOverGeneralFields 
Related topic  Perspectivity^{} 
Related topic  ProjectiveSpace 
Related topic  LinearFunction 
Related topic  Collineation 
Defines  projective transformation 
Defines  projective property 